Solve for $x$ : $ 5|x - 8| - 5 = -2|x - 8| + 8 $
Add $ {2|x - 8|} $ to both sides: $ \begin{eqnarray} 5|x - 8| - 5 &=& -2|x - 8| + 8 \\ \\ { + 2|x - 8|} && { + 2|x - 8|} \\ \\ 7|x - 8| - 5 &=& 8 \end{eqnarray} $ Add ${5}$ to both sides: $ \begin{eqnarray} 7|x - 8| - 5 &=& 8 \\ \\ { + 5} &=& { + 5} \\ \\ 7|x - 8| &=& 13 \end{eqnarray} $ Divide both sides by ${7}$ $ \dfrac{7|x - 8|} {{7}} = \dfrac{13} {{7}} $ Simplify: $ |x - 8| = \dfrac{13}{7}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 8 = -\dfrac{13}{7} $ or $ x - 8 = \dfrac{13}{7} $ Solve for the solution where $x - 8$ is negative: $ x - 8 = -\dfrac{13}{7} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& -\dfrac{13}{7} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& -\dfrac{13}{7} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $7$ $ x = - \dfrac{13}{7} {+ \dfrac{56}{7}} $ $ x = \dfrac{43}{7} $ Then calculate the solution where $x - 8$ is positive: $ x - 8 = \dfrac{13}{7} $ Add ${8}$ to both sides: $ \begin{eqnarray} x - 8 &=& \dfrac{13}{7} \\ \\ {+ 8} && {+ 8} \\ \\ x &=& \dfrac{13}{7} + 8 \end{eqnarray} $ Change the ${ + 8}$ to an equivalent fraction with a denominator of $7$ $ x = \dfrac{13}{7} {+ \dfrac{56}{7}} $ $ x = \dfrac{69}{7} $ Thus, the correct answer is $x = \dfrac{43}{7} $ or $x = \dfrac{69}{7} $.